A GENERALIZATION OF FRACTAL INTERPOLATION STOCHASTIC PROCESSES TO HIGHER DIMENSIONS

We generalize the notion of fractal interpolation functions (FIFs) to stochastic processes. We prove that the Minkowski dimension of trajectories of such interpolations for self-similar processes with stationary increments converges to 2-α. We generalize the notion of vector-valued FIFs to stochastic processes. Trajectories of such interpolations based on an equally spaced sample of size n on the interval [0,1] converge to the trajectory of the original process. Moreover, for fractional Brownian motion and, more generally, for self-similar processes with stationary increments (α-sssi) processes, upper bounds of the Minkowski dimensions of the image and the graph converge to the Hausdorff dimension of the image and the graph of the original process, respectively.